\documentclass[10pt,a4paper]{article} 

\usepackage{ctex} % 中文支持
\usepackage[top=2.5cm, bottom=2.5cm, left=2.5cm, right=2.5cm]{geometry} % 页边距
\usepackage{amsmath, amssymb} % 数学公式与符号
\usepackage{graphicx, color, url}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{titling}
\setlength{\droptitle}{-2cm} % 标题上移

\title{《基础复分析》第3章复函数 - 习题}
\author{CGZ ET AL}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\maketitle 

\begin{enumerate}

% 《基础复分析》习题三

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  % 1 

设 $f(z)$ 和 $g(z)$ 都是解析函数。证明 $f \circ g$ 也是解析函数。
    

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  % 2 

设 $f(z)$ 是解析函数, 且 $|f(z)|$ 是常数。证明 $f(z)$ 也是常数。
    

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  % 3

证明函数 $f(z)$ 和 $\overline{f(\bar{z})}$ 是同时解析的。
    

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  % 4

试给出多项式 $ax^3 + bx^2y + cxy^2 + dy^3$ 是调和函数的充分必要条件，并在此条件下求出它的共轭调和函数和对应的解析函数。
    

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  % 5 

证明调和函数 $u(z)$ 满足形式微分方程
    $$
    \frac{\partial^2 u(z)}{\partial z \partial \bar{z}} = 0.
    $$
    

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  % 6 

设 $Q(z)$ 为 $n (n \geq 1)$ 次多项式, 且具有不同的零点 $a_1, a_2, \cdots, a_n$. 设 $P(z)$ 为次数小于 $n$ 的多项式. 证明
    $$
    P(z) = \sum_{k=1}^{n} \frac{P(a_k) Q(z)}{Q'(a_k)(z - a_k)}.
    $$
    

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  % 7 

利用上题证明: 给定 $n \geq 1$ 个互不相同的复数 $a_1, a_2, \cdots, a_n$, 以及另外 $n$ 个复数 $c_1, c_2, \cdots, c_n$, 证明存在唯一的次数小于 $n$ 的多项式 $P(z)$, 使得 $P(a_k) = c_k$.
    

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  % 8 

次数 $n \geq 1$ 的有理函数的导数是有理函数。试考察它的次数。
    

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  % 9 

给出在单位圆周上绝对值为 $1$ 的有理函数的一般形式。
    

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  % 10 

求下列有理函数的临界点及其阶:
- $\dfrac{z^4}{z^3 - 2}$
- $\dfrac{1}{z^4 - 2z^2 + 3}$
- $\dfrac{z^3(2 - z)}{2z - 1}$
- $\left[\dfrac{(z^2 - 1)(z^2 + 3)}{4z^2}\right]^3$
    

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  % 11 

将下列有理函数分解为部分分式:
- $\dfrac{z^4}{z^3 - 1}$
- $\dfrac{1}{z(z + 1)(z + 2)}$
    

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  % 12 

设复数序列 $\{a_n\}$ 的极限为 $A$. 证明
    $$
    \lim_{n \to \infty} \frac{a_1 + a_2 + \cdots + a_n}{n} = A.
    $$
    

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  % 13 

讨论函数序列 $\{nz^n\}$ 的收敛性和一致收敛性。
    

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  % 14 

设 $R(z) = P(z)/Q(z)$ 为有理函数，且 $d = \deg Q - \deg P > 0$. 令
    $$
    R_n(z) = R(z)\left(1 - \frac{z}{n}\right)^d.
    $$
    证明作为 $\overline{\mathbb{C}}$ 上的连续映射序列，$\{R_n(z)\}$ 一致收敛于 $R(z)$.
    

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  % 15

设级数 $\sum a_n$ 收敛，级数 $\sum b_n$ 绝对收敛。

证明级数 $\sum c_n$ 收敛，其中
    $$
    c_n = \sum_{i=0}^{n} a_i b_{n-i}.
    $$
    

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  % 16 

如果
    $$
    \lim_{n \to \infty} \frac{|a_n|}{|a_{n+1}|} = R,
    $$
    证明 $\sum a_n z^n$ 的收敛半径为 $R$.
    

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  % 17 

求下列幂级数的收敛半径:
- $\sum n^7 z^n$
- $\sum \frac{z^n}{n!}$
- $\sum n! z^n$
- $\sum z^{n!}$
- $\sum q^{n^2} z^n (|q| < 1)$
    

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  % 18 

设 $\sum a_n z^n$ 的收敛半径为 $R$. 求 $\sum a_n z^{2n}$ 和 $\sum a_n^2 z^n$ 的收敛半径。
    

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  % 19 

设 $\sum a_n z^n$ 和 $\sum b_n z^n$ 的收敛半径分别为 $R_1$ 与 $R_2$. 证明 $\sum a_n b_n z^n$ 的收敛半径至少为 $R_1 R_2$.
    

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  % 20 

设 $\sum a_n z^n$ 的和为 $f(z)$，求 $\sum n^3 a_n z^n$ 的和。
    

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  % 21 

将 $(1-z)^{-m}$ ($m$ 为正整数) 展开为 $z$ 的幂级数。
    

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  % 22 

将 $(2z+3)/(z+1)$ 展开为 $z-1$ 的幂级数，并求收敛半径。
    

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  % 23 

考察下列级数的收敛区域:
- $$\sum_{n=0}^{\infty} \left(\frac{z}{1+z}\right)^n$$
- $$\sum_{n=0}^{\infty} \frac{z^n}{1+z^{2n}}$$
    

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  % 24 

对 $z = \pi i/2, 3\pi i/4, 2\pi i/3$, 求 $e^z$ 的值。
    

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  % 25 

求 $\exp e^z$ 的实部和虚部。


    

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  % 26 

求 $\sin i$, $\cos i$ 和 $\tan(1+i)$ 的值。
    

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  % 27 

双曲余弦和双曲正弦函数定义为
    $$
    \cosh z = \frac{e^z + e^{-z}}{2}, \quad \sinh z = \frac{e^z - e^{-z}}{2}.
    $$
    将它们用 $\cos(iz)$ 和 $\sin(iz)$ 表示出来。
    
进一步推导出加法公式，以及 $\cosh 2z$ 和 $\sinh 2z$ 的公式。
    

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  % 28 

用三角函数的加法公式将 $\cos(x+iy)$ 和 $\sin(x+iy)$ 分解为实部和虚部。
    

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  % 29 

证明
    $$
    |\cos z|^2 = \sinh^2 y + \cos^2 x = \cosh^2 y - \sin^2 x = \frac{\cosh 2y + \cos 2x}{2},
    $$
    $$
    |\sin z|^2 = \sinh^2 y + \sin^2 x = \cosh^2 y - \cos^2 x = \frac{\cosh 2y - \cos 2x}{2}.
    $$
    

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  % 30 

对 $y > 0$, 证明 $\cos y$ 的级数展开式的余项与余项的首项有相同的符号。
    

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  % 31 

证明 $$3 < \pi < 2\sqrt{3}.$$
    

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  % 32 

求复数 $2$, $-1$, $i$, $-i/2$, $-1-i$, $1+2i$ 的对数。
    

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  % 33 

记 $$\zeta = e^{2\pi i/n},\quad n \geq 2.$$ 
求 $$(1-\zeta)(1-\zeta^2)\cdots(1-\zeta^{n-1}).$$
    

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  % 34 

设 $n \geq 2$. 计算 
$$\sin \frac{\pi}{n} \sin \frac{2\pi}{n} \cdots \sin \frac{(n-1)\pi}{n}.$$





\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

